MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  abvrcl Unicode version

Theorem abvrcl 15602
Description: Reverse closure for the absolute value set. (Contributed by Mario Carneiro, 8-Sep-2014.)
Hypothesis
Ref Expression
abvf.a  |-  A  =  (AbsVal `  R )
Assertion
Ref Expression
abvrcl  |-  ( F  e.  A  ->  R  e.  Ring )

Proof of Theorem abvrcl
Dummy variables  x  y  f  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-abv 15598 . . . 4  |- AbsVal  =  ( r  e.  Ring  |->  { f  e.  ( ( 0 [,)  +oo )  ^m  ( Base `  r ) )  |  A. x  e.  ( Base `  r
) ( ( ( f `  x )  =  0  <->  x  =  ( 0g `  r ) )  /\  A. y  e.  ( Base `  r
) ( ( f `
 ( x ( .r `  r ) y ) )  =  ( ( f `  x )  x.  (
f `  y )
)  /\  ( f `  ( x ( +g  `  r ) y ) )  <_  ( (
f `  x )  +  ( f `  y ) ) ) ) } )
21dmmptss 5185 . . 3  |-  dom AbsVal  C_  Ring
3 elfvdm 5570 . . 3  |-  ( F  e.  (AbsVal `  R
)  ->  R  e.  dom AbsVal )
42, 3sseldi 3191 . 2  |-  ( F  e.  (AbsVal `  R
)  ->  R  e.  Ring )
5 abvf.a . 2  |-  A  =  (AbsVal `  R )
64, 5eleq2s 2388 1  |-  ( F  e.  A  ->  R  e.  Ring )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   {crab 2560   class class class wbr 4039   dom cdm 4705   ` cfv 5271  (class class class)co 5874    ^m cmap 6788   0cc0 8753    + caddc 8756    x. cmul 8758    +oocpnf 8880    <_ cle 8884   [,)cico 10674   Basecbs 13164   +g cplusg 13224   .rcmulr 13225   0gc0g 13416   Ringcrg 15353  AbsValcabv 15597
This theorem is referenced by:  abvfge0  15603  abveq0  15607  abvmul  15610  abvtri  15611  abv0  15612  abv1z  15613  abvneg  15615  abvsubtri  15616  abvpropd  15623  abvmet  18114  nrgrng  18190  tngnrg  18201  abvcxp  20780
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-xp 4711  df-rel 4712  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fv 5279  df-abv 15598
  Copyright terms: Public domain W3C validator